Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind

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Table 1 Exact and approximate solutions with four iterations at x = 0.5

α	Exact solution		Approximate solution		Error
α	\(\underline{u} \left( {x,\alpha } \right)\)	\(\bar{u}\left( {x,\alpha } \right)\)	\(\underline{u} \left( {x,\alpha } \right)\)	\(\bar{u}\left( {x,\alpha } \right)\)	\(\underline{E} \left( {x,\alpha } \right)\)	\(\bar{E}\left( {x,\alpha } \right)\)
0	0	4.51050	0	4.51034	0	0.00016
0.1	0.124039	4.39661	0.124034	4.39645	0.000005	0.00016
0.2	0.270630	4.27596	0.270620	4.27580	0.000010	0.00016
0.3	0.439774	4.14177	0.439758	4.14162	0.000016	0.00015
0.4	0.631471	3.98729	0.631447	3.98714	0.000024	0.00015
0.5	0.845719	3.80574	0.845688	3.80560	0.000031	0.00014
0.6	1.082520	3.59036	1.082480	3.59023	0.000040	0.00013
0.7	1.341870	3.33439	1.341820	3.33427	0.000050	0.00012
0.8	1.623780	3.03106	1.623720	3.03095	0.000060	0.00011
0.9	1.928240	2.67360	1.928170	2.67350	0.000070	0.00010
1	2.255250	2.25525	2.255170	2.25517	0.000080	0.00008